5 problems

Kozlov's conjecture. Any additional irreducible first integral of an integrable case has degree in the momenta at most $2$.

Rationality conjecture. If the equation has an elementary solution, then it is always possible to choose a first integral $I$ such that $RS$ is a rational function of $(x,y,y')$.

Formal linearization conjecture. The existence of $n-1$ functionally independent analytic strong first integrals implies that there exists a random coordinate transformation

Let $G$ be a $k$-step nilpotent Lie group with $k\geq 2$, and let its isometry Lie algebra and Lie algebra of first integrals be denoted by the corresponding Lie algebras. First-in…

Extended Prelle–Singer conjecture. If an elementary solution exists for the given second-order ordinary differential equation, then there exists at least one elementary first integ…